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IIT-JEE 2007 Paper 1 Solutions

Physics

Part ISection I

Straight Objective Type

This section contains 9 multiple choice questions numbered 1 to 9. Each question has 4 choices (A), (B),(C) and (D), out of which only one is correct.

1. A circuit is connected as shown in the figure with the switch S open. When the switch is closed,the total amount of charge that flows from Y to X is

S6 ?3 ?

9 V

Y

X3 F? 6 F?

(A) 0

(B) 54 C?

(C) 27 C?

(D) 81 C?

Sol. (C)When S is open:

3 F? 6 F?X

6?3? Y

9V After the capacitors are fully charged, current flows through resistors only.

Total charge on each capacitor is q ? ?18 F ? ?

? ?? ?? ?

q q9

3 6

When S is closed:


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3 F? 6 F?

6?3?

X

Y

9V

i i

9

i A 1A3 6

? ?? ?? ?

?? ?

Potential drop across the 3 ? resister = i 3 = 3 V

q' 3 3 c 9 c? ? ? ? ? ?

Potential drop across 6 ? resister = 6 1 V = 6V

q" 6 6 c 36 c? ? ? ? ? ?

? Net charge flowing from Y to X

? ?? ?36 18 9 ( 18) c 27 c? ? ? ? ? ? ? ? ?

2. In the options given below, let E denote the rest mass energy of a nucleus and n a neutron. Thecorrect option is

(A) ? ? ? ?236 137 97

92 53 39E U E I E Y 2E n? ? ?

(B) ? ? ? ? ? ?236 137 97

92 53 39E U E I E Y 2E n? ? ?

(C) ? ? ? ? ? ?236 140 94

92 56 36E U E Ba E Kr 2E n? ? ?

(D) ? ? ? ? ?236 140 94

92 56 36E U E Ba E Kr 2E n? ? ?

Sol. (A)

? ?? ?E parent nuclie E constituents

3. A long, hollow conducting cylinder is kept coaxially inside another long, hollow conductingcylinder of larger radius. Both the cylinders are initially electrically neutral.(A) A potential difference appears between the two cylinders when a charge density is given tothe inner cylinder(B) A potential difference appears between the two cylinders when a charge density is given tothe outer cylinder(C) No potential difference appears between the two cylinders when a uniform line charge is keptalong the axis of the cylinders(D) No potential difference appears between the two cylinders when same charge density is givento both the cylinders

Sol. (A)When a inner cylinder is charged, both negative and positive charges are induced on the outercylinder.

Using a gaussian enclosing only the inner surface, a radically symmetric electric field exists.Hence a non-zero potential difference exists.


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When outer cylinder is charged, no charges are induced on inner cylinder and hence no electricfield exists in between. Potential at all places inside the outer cylinder is constant and hence nopotential difference.

Since in (C) and (D) a net charge (line charge) exists inside a gaussian surface as chosen earlier,electric field and hence potential difference exists.

4. The largest wavelength in the ultraviolet region of the hydrogen spectrum is 122 nm. The smallestwavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is(A) 802 nm(B) 823 nm(C) 1882 nm(D) 1648 nm

Sol. (B)

2

hc 13.6 (eV)

n

??

34 82 2

19

6.6 10 3 10 100 6.6 3n n (nm)

13.6 1.613.6 1.6 10

?

?

? ? ? ? ?? ? ? ?

?? ?

291n?

(n 1) 91nm, (n 2) 364 nm? ? ? ? ? ?

(n 3) 819? ? ? nm

Note that the minimum ? of IR lies close to end of visible range. Hence (C) and (D) cannot

represent the smallest ? of IR.

Hence (B)

5. In an experiment to determine the focal length (f) of a concave mirror by the u-v method, astudent places the object pin A on the principal axis at a distance x from the pole P. The studentlooks at the pin and its inverted image from a distance keeping his/her eye in line with PA. Whenthe student shifts his/her eye towards left, the image appears to the right of the object pin. Then.(A) x < f(B) f < x < 2f(C) x = 2f(D) x > 2f

Sol. (B)From the problem statement it is clear that the image is formed behind the object. This means

v u? . Thus, f x 2f? ? .

6. A ray of light traveling in water is incident on its surface open to air. The angle of incidence is ? ,

which is less than the critical angle. Then there will be(A) only a reflected ray and no refracted ray(B) only a refraction ray and no reflected ray

(C) a reflected ray and a refracted ray and the angle between them would be less than 180 2? ?

(D) a reflected ray and a refracted ray and the angle between them would be greater than

180 2? ?


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Sol. (C)

1 2n sin n sin(90 )? ? ? ? ; 1 2n n?

?

??90 ?

n2

n1

90? ? ? ? ? ? Angle between reflected and refracted ray = 90 180 2? ? ? ? ? ? ?

7. A resistance of 2 ? is connected across one gap of a metre-bridge (the length of the wire is

100 cm) and an unknown resistance, greater than 2 ? , is connected across the other gap. When

these resistances are interchanged, the balance point shifts by 20 cm . Neglecting anycorrections, the unknown resistance is

(A) 3 ? (B) 4 ?

(C) 5 ?

(D) 6 ?

Sol. (A)

2? X

100 llo

l 100 l

2 x

??

On interchanging 2? and x,

l ' 100 l'

x 2

??

Using l' l? = 20 cm, we have

100x 20020

2 x x 2? ?

? ?

x 3? ? ?

8. Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. Thenet charge on the sphere is then,(A) negative and distributed uniformly over the surface of the sphere(B) negative and appears only at the point on the sphere closest to the point charge(C) negative and distributed non-uniformly over the entire surface of the sphere(D) zero


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Sol. (D)Although there will be a charge distribution on the outer surface of sphere, the net charge is stillzero.

q

+

+++ +

9. Two particles of mass m each are tied at the ends of a light string of length 2a. The whole systemis kept on a frictionless horizontal surface with the string held tight so that each mass is at adistance a from the center P (as shown in the figure). Now, the mid-point of the string is pulledvertically upwards with a small but constant force F. As a result, the particles move towards eachother on the surface. The magnitude of acceleration, when the separation between thembecomes 2x, is

mm

F

P

a a

(A)2 2

F a

2m a x

(B)2 2

F x

2m a x

(C)F x

2m a

(D)2 2F a x

2m x

Sol. (B)

a a

m

F

m

??

a

x

TT

F

2Tsin F? ?

xTcos ma? ?

x

F2tan

ma? ?


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? ??

x2 2

F F xa

2m tan 2m a x

yF

a2m

? , is very small and hence can be neglected.

xa a? ?

Section IIAssertion Reason Type

This section contains 4 questions numbered 10 to 13. Each question contains STATEMENT-1 (Assertion)and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is

correct.

10. Statement - 1The formula connecting u, v and f for a spherical mirror is valid only for mirrors whose sizes arevery small compared to their radii of curvature.

Because

Statement - 2Laws of reflection are strictly valid for plane surfaces, but not for large spherical surfaces(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation forstatement 1(B) Statement 1 is True, Statement 2 is True; Statement 2 is Not a correct explanation forStatement 1.(C) Statement 1 is True, Statement 2 is False(D) Statement 1 is False, Statement 2 is True

Sol. (C)

1S correct?

2S incorect?

Laws of reflection is valid for all surfaces.

11. Statement - 1A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops due tofriction between the block and surface after moving through a certain distance. The surface isnow tilted to an angle 30with the horizontal and the same block is made to go up on the surfacewith the same initial velocity v. The decrease in the mechanical energy in the second situation issmaller than that in the first situation.

Because

Statement - 2The coefficient of friction between the block and the surface decreases with the increase in theangle of inclination.(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation forstatement 1(B) Statement 1 is True, Statement 2 is True; Statement 2 is Not a correct explanation forStatement 1.(C) Statement 1 is True, Statement 2 is False(D) Statement 1 is False, Statement 2 is True


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Sol. (C)

u

V = 0

(1)

21E(1) mv2

? ? ?

u

V=0

h

(2)

21E(2) mgh mv2

? ? ?

E(2) E(1)? ? ?

1S correct? ?

2S incorrect?

? doesnt change, rather force of friction reduces.

12. Statement - 1If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic

X-rays do not change.

Because

Statement - 2When an electron beams strikes the target in an X-ray tube, part of the kinetic energy isconverted into X-ray energy.(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation forstatement 1(B) Statement 1 is True, Statement 2 is True; Statement 2 is Not a correct explanation forStatement 1.(C) Statement 1 is True, Statement 2 is False(D) Statement 1 is False, Statement 2 is True

Sol. (B)

Accelerating potential affects min? and not characteristic X-ray ?

1S correct? ?

2S correct?


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13. Statement - 1In an elastic collision between two bodies, the relative speed of the bodies after collision is equal

to the relative speed before the collision.

Because

Statement 2In an elastic collision, the linear momentum of the system is conserved.(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation forstatement 1(B) Statement 1 is True, Statement 2 is True; Statement 2 is Not a correct explanation forStatement 1.(C) Statement 1 is True, Statement 2 is False(D) Statement 1 is False, Statement 2 is True

Sol. (B) Contextually.

NB:S2 need not be true in all cases. Counter e.g. ? collision of a ball with a smooth but rigid wall.

Hence S2 ? incorrect (as a generalized statement)

Note: Elastic refers to conservation of kinetic energy only. If S2 is given as correct, S1 is generallytrue.

Section IIILinked Comprehension Type

This section contains 2 paragraphs P14-16 and P 17-19. Based upon each paragraph, 3 multiple choice

Questions have to be answered. Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONEis correct.

P14-16 : Paragraph for Questions Nos. 14 to 16.

A fixed thermally conducting cylinder has a radius R and height 0L . The cylinder is open at its bottom and

has a small hole at its top. A piston of mass M is held at a distance L from the top surface, as shown in

the figure. The atmospheric pressure is 0P

Piston

L0

L

2R


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14. The piston is now pulled out slowly and held at a distance 2 L. from the top. The pressure in thecylinder between its top and the piston will then be

(A) 0P

(B) 0P

2

(C) 02

P Mg

2 R?

?

(D) 02

P Mg

2 R?

?

Sol. (A)Since the top is open to the atmosphere, pressure inside is equal to pressure outside. Since thepiston is moved slowly, the pressure equalizes to that of atmosphere.

15. While the piston is at a distance 2L from the top, the hole at the top is sealed. The piston is then

released, to a position where it can stay in equilibrium. In this condition, the distance of the pistonfrom the top is

(A) ? ?

2

0

2

0

2P R2L

R P Mg

? ??? ?

? ?? ?

(B) ? ?

2

0

2

0

P R Mg2L

R P

? ??? ?

?? ?

(C) ? ?

2

0

2

0

P R Mg2L

R P

? ?? ?? ?

?? ?

(D) ? ?

20

2

0

P R2L

R P Mg

? ??? ?

?? ?

Sol. (D)

y

2R

Since the cylinder is thermally conducting, the expansion is isothermal. Let P be the pressureinside the cylinder at equilibrium.

2 20P (2L) R P(y) R? ? ?

02P LP

y

? ?

Equilibrium ? Forces are balanced.

?2 20P R Mg P R? ? ? ? ?

00 2

2 P L MgP

y R? ?

?

20

20

2P R Ly

R P Mg

?? ?

? ?


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16. The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank isbrought below the cylinder and put in a position so that the water surface in the tank is at the

same level as the top of the cylinder as shown in the figure. The density of water is ? . In

equilibrium, the height H of water column in the cylinder satisfies

L0

H

(A) ? ? ? ?

2

o 0 o 0 0g L H P L H L P 0? ? ? ?

(B) ? ? ? ?2

o 0 o 0 0g L H P L H L P 0? ? ? ?

(C) ? ? ? ?2

o 0 o 0 0g L H P L H L P 0? ? ? ?

(D) ? ? ? ?2

o 0 o 0 0g L H P L H L P 0? ? ? ?

Sol. (C)Let the air pressure be P (inside the cylinder). Then

? ? ? ?2 2

0 0 0P L H R P L R? ? ? ?

00

0

LP P

L H

? ?? ? ? ?

?? ?

This pressure is balanced by the hydrostatic pressure. So

? ?0

0 0 00

LP g L H P

L H

? ?? ? ? ?? ?

?? ?

? ? ? ?2

0 0 0 0 0g L H P L H P L 0? ? ? ? ? ? ?

P17-19 : Paragraph for Questions Nos. 17 to 19

Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I

respectively about the common axis. Disc A is imparted an initial angular velocity 2? using the entire

potential energy of a spring compressed by a distance1x . Disc B is imparted an angular velocity ? by a

spring having the same spring constant and compressed by a distance2x . Both the discs rotate in the

clockwise direction

17. The ratio 1 2x / x is

(A) 2

(B)1

2

(C) 2

(D)1

2


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Sol. (C)

? ?

22 21

1 1kx I 2 2I2 2? ? ? ?

221

4Ix

k

?? ?

Also, ? ?2 2 22

1 1kx 2I I

2 2? ? ? ?

222

2Ix

k

?? ?

1

2

x2

x? ?

18. When disc B is brought in contact with disc A, they acquire a common angular velocity in time t.The average frictional torque on one disc by the other during this period is

(A)2I

3t

?

(B)9I

2t

?

(C)9I

4t

?

(D)3I

2t

?

Sol. (A)

If the common angular velocity is 0? ,

? ? ? 0I 2 2I 3I? ? ? ? ?

04

3? ? ? ?

Now, for disc A,

0 2 t? ? ? ? ?

02 2

t 3t

? ? ? ?? ? ? ?

2II

3t

?? ? ? ? ?

19. The loss of kinetic energy during the above process is

(A)2I

2

?

(B)2I

3

?

(C)2I

4

?

(D)2I

6

?


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Sol. (B)

? ? ? ?

22i

1 1KE I 2 2I2 2? ? ? ? 23I? ?

? ?

22

f1 4 8

KE 3I I2 3 3

?? ?? ? ?? ?

? ?

? Loss of KE =2I

3

?

Section IVMatrix-Match Type

This section contains 3 questions . Each question contains statement given in two columns which have tobe matched. Statement (A, B, C, D) in column I have to be matched with statements (p, q, r, s) incolumn II. The answers to these questions have to be appropriately bubbled as illustrated in the followingexample.

If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s then the correct bubbled 4 4 matrixshould be as follows

p q r s

p q r s

p q r s

p q r s

A

B

C

D

p q r s

20. Some physical quantities are given in column I and some possible SI units in which thesequantities may be expressed are given in column II. Match the physical quantities in column Iwith the units in Column II and indicate your answer by darkening appropriate bubbles in the 4

4 matrix given in the ORS.

Column I Column II

(A)

e sGM M

G universal gravitational constant,

eM mass of the earth

sM Mass of the Sun

(p) (volt) (coulomb) (metre)

(B)

3RT

M

R universal gas constantT absolute temperatureM molar mass

(q) ? ?? ? ? ?3 2

ki logram metre second

(C)

2

2 2

Fq B

F forceQ chargeB magnetic field

(r) ? ? ? ?2 2

metre sec ond

(D)

e

e

GM

R

G universal gravitational constant,

eM mass of the earth

eR radius of the earth

(S) ? ?? ? ? ?2 1

farad volt kg


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Sol.(A) ? (p), (q)

(B) ? (r), (s)

(C) ? (r) , (s)

(D) ? (r), (s)

21. Column I gives certain situations in which a straight metallic wire of resistance R is used andcolumn II gives some resulting effects. Match the statement in column I with the statements incolumn II and indicate your answer by darkening appropriate bubbles in the 4 4 matrix given inthe ORS.

Column I Column II

(A)A charged capacitor is connected tothe ends of the wire

(p) A constant current flows through the wire

(B)

The wire is moved perpendicular toits length with a constant velocity ina uniform magnetic fieldperpendicular to the plane of motion

(q) Thermal energy is generated in the wire

(C)The wire is placed in a constantelectric field that has direction alongthe length of the wire

(r)A constant potential difference developsbetween the ends of the wire

(D)A battery of constant emf isconnected to the ends of the wire

(S)Charges of constant magnitude appearat the ends of the wire

Sol.

(A) ? (q)

(B) ? (r), (s),

(C) ? (s)

(D) ? (p), (q) , (r)

22. Some laws/ processes are given in Column I. Match these with physical phenomena given inColumn II and indicate your answer by darkening appropriate bubbles in the 4 4 matrix given inthe ORS.

Column I Column II

(A)Transition between two atomicenergy levels

(p) Characteristic X-rays

(B) Electron emission from a material (q) Photoelectric effect(C) Mosleys law (r) Hydrogen spectrum

(D) Change of photon energy intokinetic energy of electrons (S) decay?

Sol.

(A) ? (p), (r)

(B) ? (q), (s),

(C) ? (p)

(D) ? (q)


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Chemistry

Part II

Useful data:

Gas Contant, R = 8.314 K 1 1JK mol

1 F = 96500 C

Atomic Numbers:H = 1, Li = 3, B= 5, C = 6, N = 7, O = 8, F = 9, Na = 11, P = 15, S = 16, Cl = 17, Ar = 18,K = 19, V = 23, Cr = 24, Mn = 25, Fe = 26, Co= 27, Ni = 28,Cu = 29, Zn = 30, Ge = 32, Br = 35, Ag = 47,I = 53, Xe= 54, Pt = 78, Hg = 90 , Pb = 82

Section IStraight Objective Type

This section contains 9 multiple choice questions numbered 23 to 31. Each question has 4 choices (A),(B), and (D), out of which only one is correct

23. The number of structural isomers for 6 14C H is

(A) 3(B) 4(C) 5(D) 6

Sol. (C)Structural isomers of C6H14

3 2 2 2 2 3CH CH CH CH CH CH

3 2 2 3|

CH3

CH CH CH CH CH

3 2 2 3|

CH3

CH CH CH CH CH

3 3| |

CH CH3 3

CH CH CH CH

CH CCH CH3 2 3

CH3

CH3

24. When 20 g of naphthoic acid 11 8 2C H O is dissolved in 50 g of benzene ?1

fK 1.72 K kg mol? , a

freezing point depression of 2 K is observed. The vant Hoff factor (i) is(A) 0.5(B) 1(C) 2(D) 3


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Sol. (A)

? ? ?? ? ?

? ?

fT 2 172 50i 0.5

Km 1.72 20 2000

25. In the following reaction,

O

NH

conc. HNO3conc. H SO2 4

X

the structure of the major product X is

(A)

NH

O

NO2

(B)

NH

OO N2

(C)

NH

O

NO2

(D)

NH

O

O2N

Sol. (B)O N2

N

H

C

O

BecauseNH

group is activator and p-directing towards electrophilic aromatic substitution.


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26. The value of 10log K for a reaction A B is

(Given ? ? ? ?o 1 o 1 1

r 298K r 298KH 54.07 kJmol , S 10 JK mol and 1R 9314JK mol ;2.303 8.314 298 5705)? ? ? ?

(A) 5(B) 10(C) 95(D) 100

Sol. (B)

? ? ?G 2.303 RTlog K

G H T S? ? ? ? ?

54070 2980? ? ?

54070 J? ?

? 54070 = 2.303 8.314 298 log Klog K = 10

27. Among the following, the paramagnetic compound is

(A)2 2Na O

(B)3O

(C) 2N O

(D)2KO

Sol. (D)

2KO is paramagnetic is nature because it contains unpaired electrons.

28. The percentage of p-character in the orbitals forming PP bonds in 4P is

(A) 25(B) 33(C) 50(D) 75

Sol. (D)

Phosphorous will show 3sp hybridisation having 75% p-character.

P P

P

P

29. The species having bond order different from that in CO is

(A) NO

(B) NO?

(C) CN

(D)2N


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Sol. (A)

Bond order of NO? is 2 while bond order of CO is 3.

30. The reagent(s) for the following conversion,Br

Br H H?

is/are(A) alcoholic KOH

(B) alcoholic KOH followed by 2NaHH

(C) aqueous KOH followed by 2NaNH

(D) 3Zn/CH OH

Sol. (B)

Br

CH2

CH2

Br

alc. KOH/NaNH2H C C H

31. Extraction of zinc form zinc blende is achieved by(A) electrolytic reduction(B) roasting followed by reduction with carbon(C) roasting followed by reduction with another metal(D) roasting followed by self-reduction

Sol. (B)Roasting followed by reduction with carbon.

Section II

Assertion Reason Type

This section contains 4 questions numbered 32 to 35. Each question contains STATEMENT-1 (Assertion)and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which only one iscorrect.

32. Statement - 1p-Hydroxybenzoic acid has a lower boiling point than o-hydroxybenzoic acid

Because

Statement - 2o-Hydroxybenzoic acid has intramolecular hydrogen bonding

(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation forstatement 1(B) Statement 1 is True, Statement 2 is True; Statement 2 is Not a correct explanation forStatement 1.(C) Statement 1 is True, Statement 2 is False(D) Statement 1 is False, Statement 2 is True

Sol. (D)O-hydroxybenzoic acid has intramolecular hydrogen bonding hence has lower boiling point thanp-hydroxybenzoic acid.


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33. Statement - 1Boron always forms covalent bond

Because

Statement - 2

The small size of 3B ? favours formation of covalent bond.


Sol. (A)

Boron always forms covalent bond, because its compound will have zero resultant dipole momentdue to its trigonal plannar shape.

34. Statement - 1Micelles are formed by surfactant molecules above the critical micellar concentration (CMC)

Because

Statement - 2The conductivity of a solution having surfactant molecules decreases sharply at the CMC.

(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation forstatement 1

(B) Statement 1 is True, Statement 2 is True; Statement 2 is Not a correct explanation forStatement 1.(C) Statement 1 is True, Statement 2 is False(D) Statement 1 is False, Statement 2 is True

Sol. (B)

35. Statement - 1In water, orthoboric acid behaves as a weak acid monobasic acid.

Because

Statement 2

In water, orthoboric acid acts as a proton donor.


Sol. (C)


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This section contains 2 paragraphs C36-38 and C39-41. Based upon each paragraph, 3 multiple choiceQuestions have to be answered. Each question has 4 choices (A), (B), (C) and (D) out of which Only oneis correct.


Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules

(approximately 6.023 2310 ) are present in a few grams of any chemical compound varying with their

atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced.This concept has implications in diverse areas such as analytical chemistry, biochemistryelectrochemistry and radiochemistry. The following example illustrates a typical case, involving

chemical/electrochemical reaction, which requires a clear understanding of the mole concept.

A 4.0 molar aqueous solution NaCl is prepared and 500 mL of this solution is electrolysed. This leads tothe evolution of chlorine gas at one of the electrodes (atomic mass : Na = 23, Hg = 200;1 Faraday =96500 coulombs).

36. The total number of moles of chlorine gas evolved is(A) 0.5(B) 1.0(C) 2.0(D) 3.0

Sol. (B)

37. If the cathode is a Hg electrode, the maximum weight (g) of amalgam formed from this solution is(A) 200(B) 225(C) 400(D) 446

Sol. (D)In presence of amalgam cathode sodium ion will discharge in place of hydrogen gas due to overvoltage is the form of amalgams.Weight of amalgam = 2 (23 + 200) = 446 g

38. The total charge (coulombs) required for compete electrolysis is(A) 24125(B) 48250(C) 96500(D) 193000

Sol. (D)Total charge = 2 96500 = 193000 C


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The noble gases have closed-shell electronic configuration and are monoatomic gases under normal

conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces betweenthe atoms and the absence of other interatomic interactions.

The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers +2, +4

and + 6. 4XeF reacts violently with water to give 3XeO . The compounds of xenon exhibit rich

stereochemistry and their geometries can be deduced considering the total number of electron pairs inthe valence shell

39. Argon is used in arc welding because of its(A) low reactivity with metal(B) ability to lower the melting point of metal(C) flammability(D) high calorific value

Sol. (A)

Argon may be used to provide an inert atmosphere in welding.

40. The structure of 3XeO is

(A) linear(B) planar(C) pyramidal(D) T-shaped

Sol. (C)Pyramidal

41. 4XeF and 6XeF are expected to be

(A) oxidizing(B) reducing(C) unreactive(D) strong basic

Sol. (A)


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p q r s

p q r s

p q r s

p q r s

A

B

C

D

p q r s

42. Match the complexes in column I with their properties listed in column II. Indicate your answerby darkening the appropriate bubbles of the 4 4 matrix given in the ORS.

Column I Column II

(A) ? ?3 2 24 2Co NH H O Cl? ?? ? (p) geometrical isomers

(B) 3 22Pt NH Cl? ?? ? (q) paramagnetic

(C) 2 5Co H O Cl Cl? ?? ? (r) diamagnetic

(D) 2 26Ni H O Cl Cl? ?? ? (s) metal ion with +2 oxidation state

Sol.A p, A q, A s.B p, B s, B rC s, C qD s, D q

43. Match the chemical substances in column I with type of polymers/type of bonds in column II.Indicate your answer by darkening the appropriate bubbles of the 4 4 matrix given in the ORS.

Column I Column II(A) Cellulose (p) natural polymer(B) nylon-6,6 (q) synthetic polymer(C) Protein (r) amide linkage(D) Sucrose (s) glycoside linkage


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Sol.A p, A sB q, B r

C r, C pD s

44. Match gases under specified conditions listed in column I with their properties/laws in column II.Indicate your answer by darkening the appropriate bubbles of the 4 4 matrix given in the ORS.

Column I Column II

(A)hydrogen gas (P = 200 atm,T = 273 K)

(p) compressibility factor ? 1

(B) hydrogen gas (P = 0, T = 273 K) (q) attractive forces are dominant

(C) 2CO (P = 1 atm, T = 273 K) (r) PV = nRT

(D)real gas with very large molar

volume

(s) P(V nb) = nRT

Sol.A p, A s, B r, C q, C p, D p, D q


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Mathematics

Section IIIStraight Objective Type

This section contains 9 multiple choice questions numbered 45 to 53. Each question has 4 choices (A),(B), (C) and (D), out of which only one is correct

45. A hyperbola, having the transverse axis of length 2sin ? , is confocal with the ellipse2 23x 4y 12? ? . Then its equation is

(A) 2 2 2 2x cosec y sec 1? ? ? ?

(B) 2 2 2 2x sec y cosec 1? ? ? ?

(C) 2 2 2 2x sin y cos 1? ? ? ?

(D) 2 2 2 2x cos y sin 1? ? ? ?

Sol. (A)2 2 2 2x cosec y sec 1? ? ? ?

a sin? ? for hyperbola

? ? ?ae 4 3 12 2 2 2b sin (cosec 1) cos? ? ? ? ? ?

2 2

2 2

x y1

sin cos? ?

? ?

2 2 2 2x cosec y sec 1? ? ? ?

46. A man walks a distance of 3 units from the origin towards the north-east (N 45 E) direction. Fromthere, he walks a distance of 4 units towards the north-west (N 45W) direction to reach a pointP. Then the position of P in the Argand plane is

(A) i / 43e 4i? ?

(B) ?i / 43 4i e ?

(C) ?i / 44 3i e ??

(D) ?i / 43 4i e ??

Sol. (D)

? ? i / 43 4i e?

?

45

3

0

Z

41i tan

4 3Z 5.e

?? ???? ?

? ??


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41itani / 435.e .e

?

??

i / 43 45 i .e5 5

?? ?? ?? ?? ?

? ?i / 43 4i e ?? ?

47. The tangent to the curve xy e? drawn at the point ?cc, e intersects the line joining the points

?c 1c 1, e ?? and ?

c 1c 1, e ??

(A) on the left of x = c(B) on the right of x = c(C) at no point(D) at all points

Sol. (A)On the left of x = c

Shortcut: Let c = 0

48. The number of solutions of the pair of equations22sin cos2 0? ? ? ? 2

2cos 3sin 0? ? ? ? in the interval 0, 2? is

(A) zero(B) one(C) two(D) four

Sol. (C)Add two equations

22 1 2sin 3sin 0? ? ? ? ? ? 22sin 3sin 1 0? ? ? ? ?

? ?2sin 1 sin 1 0? ? ? ? ?

1sin , 1

2? ?

If sin 1? ? , cos 0? ? not satisfies second equation.

? ? ? ? ? ? ?1 3

sin cos2 2

5,

6 6

? ?? ?


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49. Let f(x) be differentiable on the interval ?0, ? such that f(1) = 1, and

2 2

t x

t f(x) x f(t)lim 1t x?

?

??

for each x > 0. Then f(x) is

(A)21 2x

3x 3?

(B)21 4x

3x 3

??

(C)2

1 2

x x

??

(D)1

x

Sol. (A)21 2x

3x 3?

?

?? ?

?

2 2

t x

t f(x) x f(t)lim 1

t x

0

0

? ?? ?? ?

?

?? ?? ?

2

t x

x f(x) x f (t)lim 1

1

?? ? ?22xf(x) x f (x) 1

2

dy 2 1y

dx x x

?? ?

dx2 2lnx 2xI.F. e e x

? ???

? ? ?

Solution is given by

?? ? ??

2

4

1yx dx c

x

3xc

3

?

? ?

at x = 1, y = 12

c3

? ?

50. The number of distinct real values of ? ,for which the vectors 2 i j k? ? ? ? , 2 i j k? ? ? and

2 i j k? ? ? are coplanar, is

(A) zero(B) one(C) two(D) three


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Sol. (C)

2

2

2

1 11 1 0

1 1

? ?

? ? ?

? ?

?2

2 22 1 0? ? ? ? ? ? ? 2? ? ? ?

51. Let ? , ? be the roots of the equation 2x px r 0? ? ? and2

?, 2? be the roots of the equation

2x qx r 0? ? ? . Then the value of r is

(A) ? ?? ?2

p q 2q p9

? ?

(B) ? ?? ?2 q p 2p q9? ?

(C) ? ?? ?2

q 2p 2q p9

? ?

(D) ? ?? ?2

2p q 2q p9

? ?

Sol. (D)

? ?? ?2

2p q 2q p9

? ?

p? ? ? ?

r? ? ?

2 q2

?? ? ?

4 2q? ? ? ?

2q p

3

?? ?

4p 2q

3

?? ? ; ? ?? ?

2r 2q p 2p q

9? ? ?

52.

2sec x

2

2

x 24

f(t)dt

lim

x16

?

??

?

?equals

(A)8

f(2)?

(B)2

f(2)?

(C)2 1

f2

? ?? ?

? ? ?

(D) 4f(2)


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Sol. (A)

? ?? ?

?

??

?

2sec x

2 22

22x x

4 4

f(t)dt2f(sec x )sec x tanx

lim lim2x

x16

2f(2).2.1 8f(2)

2.4

? ?? ?

53. One Indian and four American men and their wives are to be seated randomly around a circulartable. Then the conditional probability that the Indian man is seated adjacent to his wife given thateach American man is seated adjacent to his wife is

(A)1

2

(B)1

3

(C)2

5

(D)1

5

Sol. (C)

? ? ? ? ?1 1 2 2 3 3 4 4A A A A A A A A II

Total 45! 2?

? ? ? ? ?1 1 2 22 3 4 43A A A A A A A A II

Favourable 54! 2? 5

4

4! 2 2

55! 2

??

?

Section IIAssertion Reason Type

This section contains 4 questions numbered 54 to 57. Each question contains STATEMENT-1 (Assertion)

and STATEMENT-2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which only one iscorrect.


28/34


54. Let the vectors PQ, QR, RS, ST, TU and UP represent the sides of a regular hexagon.

Statement -1:PQ RS ST 0? ? ?

Because

Statement - 2

PQ RS 0 and PQ ST 0? ? ? ?


Sol. (C)

1S ? True, 2S ? False

RS ST RT? ?

PQ RT 0? ? as PQ RT?

T S

R

Q

V

P

PQ RS 0? ?

55. Let1 2 nH , H ,...., H be mutually exclusive and exhaustive events with iP(H ) 0? , i 1, 2, . ..,n? . Let E

be any other event with 0 P(E) 1? ? .

Statement - 1

i iP(H | E) P(E | H )? . iP(H ) for i = 1, 2,,n.

Because

Statement - 2n

ii 1

P(H ) 1?

??



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Sol. (D)

1S ? False, 2S ? True ,

? ? ? ?? iH EP Hi / E PP(E)

? ? ? ??

? ? ? ?i

i i ii

P E HP E / Hi P(H ) .P(H ) P E H

P(H

? ?? iP Hi / E P E / Hi .P(H )

as P(E) 1?

56. Let F(x) be an indefinite integral of 2sin x .

Statement - 1

The function F(x) satisfies F(x ) F(x)? ? ? for all real x.

Because

Statement - 22 2sin (x ) sin x? ? ? for all real x.


Sol. (D)

1S False?

x sin2xf(x) c

2 4? ? ?

57. Tangents are drawn from the point (17, 7) to the circle 2x y 169?? ?

Statement - 1

The tangents are mutually perpendicular.

Because

Statement - 2

The locus of the points from which mutually perpendicular tangents can be drawn to the givencircle is 2x y 338?? ? .



30/34


Sol. (A)

1 2S True, S True? ? , 2S explains 1S

Director circle of2 2 2

x y r? ? is2 2 2

x y 2r? ?


This section contains 2 paragraphs M58-60 and M61-63. Based upon each paragraph, 3 multiple choiceQuestions have to be answered. Each question has 4 choices (A), (B), (C) and (D) out of which Only oneis correct.

M58-60 : Paragraph for Questions Nos. 58 to 60.

Consider the circle ?? ?2x y 9 and the parabola y 8x? ? .They intersect at P and Q in the first and the

fourth quadrants, respectively.Tangents to the circle at P and Q intersect the x-axis at R and tangents tothe parabola at P and Q intersect the x-axis at S.

58. The ratio of the areas of the triangles PQS and PQR is

(A) 1: 2

(B) 1 : 2(C) 1 : 4(D) 1 : 8

Sol. (C)

RS

( )P 1, 2 2

( )Q 1, 2 2

? ?PR : x 2 2y 9

? ?QR : x 2 2y 9

?R 9, 0?

?? ?PS : 2 2y 4 x 1

? ?? ? ?QS : 2 2y 4 x 1 S( 1, 0)? ?

14 2 2

PQS 2 1: 41PQR

4 2 82

? ??

? ??

? ?


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59. The radius of the circumcircle of the triangle PRS is(A) 5

(B) 3 3

(C) 3 2

(D) 2 3

Sol. (B)

abcR

4?

?

60. The radius of the incircle of the triangle PQR is(A) 4(B) 3

(C)8

3

(D) 2

Sol. (D)

rs

??

M61-63 : Paragraph for Questions Nos. 61 to 63.

Let rV denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the

common difference is 2r 1? .

Let r r 1 r r r 1 rT V V 2 and Q T T? ?? ? ? ? ? for r = 1, 2,.

61. The sum 1 2 nV V ... V? ? ? is

(A) 21

n(n 1)(3n n 1)12

? ? ?

(B) 21

n(n 1)(3n n 2)12

? ? ?

(C) 21

n(2n n 1)2

? ?

(D) 31

n(2n 2n 3)3

? ?

Sol. (B)2

2r

r rV r

2 2? ? ? ; ? ?

2r

1V n n 1 3n n 2

12? ? ? ? ?

62. rT is always

(A) an odd number(B) an even number(C) a prime number(D) a composite number


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Sol. (D)Always composite number

? ?? ? ?r

T 3r 1 r 1

63. Which one of the following is a correct statement?

(A)1 2 3Q , Q , Q ,... are in A.P. with common difference 5

(B) 1 2 3Q , Q , Q ,... are in A.P. with common difference 6

(C) 1 2 3Q , Q , Q ,... are in A.P. with common difference 11

(D)1 2 3Q Q Q ...? ? ?

Sol. (B)

1 2 3Q , Q , Q ..... are in A.P. with common difference 6

rQ 6r 5? ?




p q r sp q r s

p q r s

p q r s

AB

C

D

p q r s

64. Consider the following linear equations

ax by cz 0? ? ?

bx cy az 0? ? ?

cx ay bz 0? ? ?

Match the conditions/expressions in Column I with statement in Column II and indicate youranswer by darkening the appropriate bubbles in the 4 4 matrix given in the ORS.

Column I Column II

(A)a b c 0 and? ? ?

2 2 2a b c ab bc ca? ? ? ? ? (p)

the equations represent planes meetingonly at a single point.

(B)a b c 0 and? ? ?

2 2 2a b c ab bc ca? ? ? ? ? (q) the equations represent the line x = y =z

(C)a b c 0 and? ? ?

2 2 2a b c ab bc ca? ? ? ? ? (r) the equations represent identical planes.

(D)a b c 0 and? ? ?

2 2 2a b c ab bc ca? ? ? ? ? (s)

the equations represent the whole of thethree dimensional space.


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Sol.A ? r, B ? q, C ? p, D ? s

65. In the following ?x denotes the greatest integer less than or equal to x.

Math the functions in Column I with the properties in Column II and indicate your answer bydarkening the appropriate bubbles in the 4 4 matrix given in the ORS

Column I Column II

(A) x x (p) continuous in (1, 1)

(B) x (q) differentiable in (1, 1)

(C) x x? (r) strictly increasing in (1, 1)

(D) x 1 x 1? ? ? (s)not differentiable at least at one point in(1, 1)

Sol.

A p, q, r?

1 1

x x ? continuous in (1, 1) differentiable in (1, 1) strictly increasing in (1, 1)

(B) ? p, s

1 1 0

x

x

x ? continuous in (1, 1) and differentiable in (1, 1)

(C) ? r, s

1 1

?x x? ? strictly increasing in (1, 1) not differentiable atleast at one point in (1, 1)

(D) ? p, q


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2

2x 2x

1 1 x 1 x 1? ? ? ? continuous in (1, 1) and differentiable in (1, 1)

?

?

?

?

A p, q, r

B p, s

C r, s

D p, q

66. Match the integrals in Column I with the values in Column II and indicate your answer bydarkening the appropriate bubbles in the 4 4 matrix given in the CRS.

Column I Column II

(A)1

2

1

dx

1 x?

?? (p)

1 2log

2 3

? ?? ?? ?

(B)1

20

dx

1 x?? (q)

22log

3

? ?? ?? ?

(C)

3

2

2

dx

1 x?? (r)

3

?

(D)

2

21

dx

x 1 x?? (s)

2

?

Sol.A s?

111

2

1 1

dxtan x

21 x

?

? ?

?? ?

??

B s? 11

1

20 0

dxsin x

21 x

? ?? ?

??

C p?

3 3

e22

2

dx 1 1 xlog2 1 x1 x

??

???

e1 2

log2 3

?

D r?

22

1 1

2 11

dxsec x sec 2

3x x 1

? ? ?? ? ?

??

iit-jee – 2007 paper – 1 solutions

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